Find the vertex , focus , latusrectum, axis and the directrix of the parabola `x^2+8x+12y+4=0`.

Answer

To solve the problem step by step, we will start with the given equation of the parabola and then find the vertex, focus, latus rectum, axis, and directrix.

### Step 1: Rewrite the equation
The given equation is:
\[ x^2 + 8x + 12y + 4 = 0 \]

We can rearrange this equation:
\[ x^2 + 8x = -12y - 4 \]

### Step 2: Complete the square
To complete the square for the left side:
1. Take the coefficient of \( x \) which is 8, halve it to get 4, and then square it to get 16.
2. Add and subtract 16 on the left side.

Thus, we rewrite:
\[ x^2 + 8x + 16 - 16 = -12y - 4 \]
\[ (x + 4)^2 - 16 = -12y - 4 \]
\[ (x + 4)^2 = -12y + 12 \]
\[ (x + 4)^2 = -12(y - 1) \]

### Step 3: Identify the standard form
The equation is now in the standard form of a parabola:
\[ (x - h)^2 = 4p(y - k) \]
where \( (h, k) \) is the vertex and \( 4p \) is the coefficient of \( (y - k) \).

From our equation:
- \( h = -4 \)
- \( k = 1 \)
- \( 4p = -12 \) which gives \( p = -3 \)

### Step 4: Find the vertex
The vertex \( (h, k) \) is:
\[ \text{Vertex} = (-4, 1) \]

### Step 5: Find the focus
The focus is located at:
\[ (h, k + p) = (-4, 1 - 3) = (-4, -2) \]

### Step 6: Find the directrix
The directrix is given by the equation:
\[ y = k - p = 1 + 3 = 4 \]

### Step 7: Find the latus rectum
The length of the latus rectum is given by \( |4p| \):
\[ |4p| = 12 \]

### Step 8: Identify the axis of the parabola
The axis of the parabola is the line that passes through the vertex and is parallel to the axis of symmetry. Since this parabola opens downwards, the axis is vertical:
\[ x = -4 \]

### Summary of Results
- **Vertex**: \( (-4, 1) \)
- **Focus**: \( (-4, -2) \)
- **Directrix**: \( y = 4 \)
- **Latus Rectum**: Length = 12
- **Axis**: \( x = -4 \)